A Fair Policy for the Servers in the G / GI / N Queue Josh Reed NYU Stern School of Business Joint work with Yair Shaki Stochastic Networks Conference June 20, 2012 Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 1 / 46
Introduction Modern call centers employ 100’s, if not 1000’s of agents. Moreover, often times agents may have distinct skill sets. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 2 / 46
Introduction For example, some of the agents staffed in a call center at a bank may be very good at opening a new account while others may be more skilled in handling cases of fraud. Some agents may even be trained to handle both of these types of customer service requests. Some agents might be fast at processing requests, while others might be slower. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 3 / 46
Introduction A natural question which arises in these settings is how to decide who to route incoming customers to? Assume that there is a single customer class with several heterogeneous servers. Always route to the fastest available server? Perhaps use some sort of threshold rule? What should the proper objective be? Minimizing customer waiting times sounds reasonable but may not always be the best choice. Why not? Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 4 / 46
Introduction Minimizing customer waiting times might be good for the customers arriving to the system but not so great for the servers themselves. This is especially true if the servers are human beings (as opposed to machines) as is the case for a telephone call center. In this talk, we will look how to develop policies which are efficient from the customers’ point of view but are also “fair” to the servers in the system. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 5 / 46
Outline of the Talk Introduction Literature Review Asymptotic Regime u -Greedy Policies Main Results Proof Techniques Future Work Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 6 / 46
Literature Review In (1984), Lin and Kumar considered the following system. µ 1 λ µ 1 > µ 2 µ 2 There is only a single buffer and the decision is when a server becomes free, should you send a customer to it or not? Clearly, you should always send a customer to the faster server. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 7 / 46
Literature Review But what about the slower server? Lin and Kumar proved that in order to minimize customer sojourn times you should only route to the slower server when the number of customers in the buffer is above a certain threshold. More difficult find solutions to when there are more than two servers. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 8 / 46
The Inverted-V Model λ µ 1 µ 2 µ L This is sometimes referred to as the “inverted-V” model. Because of the difficulty of handling multiple types of servers, many authors have turned to an asymptotic analysis. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 9 / 46
The Asymptotic Regime Consider a sequence of systems indexed by the number of servers N which we let tend to ∞ . The system with N servers has an arrival rate of λ N . Each system has L ≥ 1 server pools (fixed, does not change with N ) and the number of servers in server pool l for l = 1 , ..., L , is given by N l = ⌊ ν l N ⌋ , where ν 1 + ... + ν L = 1 . Service times in server pool l have a fixed distribution F l with mean 1 /µ l . Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 10 / 46
The Asymptotic Regime The capacity of the system with N servers is approximately L � ⌊ ν l N l ⌋ µ l . l =1 In the Halfin and Whitt regime, we assume that capacity is approximately matched with the incoming demand rate. In particular, letting � L � 1 λ N − β N � = √ ⌊ ν l N ⌋ µ l , N l =1 we assume that β N → β < 0 as N → ∞ . Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 11 / 46
The Asymptotic Regime The previous convergence implies that L √ √ λ N − β � ⌊ ν l N ⌋ µ l = N + o ( N ) . l =1 In other words, the capacity of the system differs from the arrival rate √ by a O ( N ) term. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 12 / 46
Literature Review Armony (2005) was one of the first authors to consider the inverted-V model in the Halfin and Whitt asymptotic regime. She considered the fastest server first (FSF) routing policy. Incoming customers are routed to the fastest available server. If it happens to be the slowest server in the system, no problem. Armony showed that for exponentially distributed service time in each server pool, FSF asymptotically minimizes customer waiting times. In particular, no thresholds are needed. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 13 / 46
Literature Review 100% Perecent of Servers Busy Fast Servers Slow Servers Time Unfortunately, under the FSF routing policy, the slow servers will be √ given O (1 / N ) idle time, while the fast server pool will never be allowed to idle. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 14 / 46
Literature Review Tezcan (2011) considered H ∗ 2 service time distributions which are a mixture of an exponential and a point mass at zero. Showed that a static priority policy is optimal but not necessarily FSF. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 15 / 46
Literature Review Atar (2008) considered the longest idled served first (LISF) routing policy for the inverted-V model. LISF tends to be biased towards fast servers. They will finish serving customers more often and so will end up having longer cumulative idle times. Gurvich and Whitt propose (2009) Idleness Ratio (IR) routing which attempts to keeps the idle servers in fixed proportions. Performs similar to LISF asymptotically. Mandelbaum, Momcilovic and Tseytlin (2012) consider the Randomized Most-Idle (RMI) policy which uniformly at random picks an available server to route to next. Also performs similar to LISF asymptotically. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 16 / 46
Literature Review Armony and Ward (2010) proposed a middle ground between minimizing customer waiting times and achieving server fairness. Their objective is to minimize customer waiting times subject to the steady state percentage of idle servers from each server pool being fixed constants. In Armony and Ward, it is shown that the asymptotically optimal policy is a FSF-excluding-pool- k policy. This policy operates similar to the FSF policy with the exception that server pool k is given the lowest priority. The server pool k varies depending on the number of customers in the system. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 17 / 46
Literature Review 0.021 1.0000 FSF 0.9000 0.020 Threshold Slow Server Idleness E[Wait per Unit Time] 0.8000 LISF 0.019 0.7000 Proportion 0.6000 0.018 0.5000 0.017 0.4000 0.3000 0.016 0.2000 FSF 0.015 Threshold 0.1000 LISF 0.0000 0.014 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2 Fast Server Rate Fast Server Rate Figure : from Armony and Ward (2010) Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 18 / 46
Literature Review In (2011), Atar, Shaki and Shwartz modified LISF to longest cumulative idled served first (LIPF). This policy routes customers to the server pool with the longest cumulative idleness. Atar, Shaki and Shwartz showed that longest cumulative idled served first asymptotically equalizes cumulative idleness. They considered exponentially distributed service times in each server pool. However, empirical evidence suggests that service times at telephone call centers are not exponentially distributed. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 19 / 46
Literature Review Figure : Picture from Brown et. al (2005) Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 20 / 46
Main Results Our goal in this work is to extend the results of Atar, Shaki and Shwartz to general service time distributions. In the process, we develop a new technique for the asymptotic analysis of many server queues with general service time distributions. This technique is based off of a simple conservation of flow identity and appears to be promising for analyzing a wide variety of routing policies for the inverted-V model such as FSF or LISF. It can also be used in the analysis of networks of many server queues. Josh Reed (NYU) A Fair Policy for the Servers in the G / GI / N Queue June 20, 2012 21 / 46
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